Question

How do you find the inverse of `f(x)=e^-x`?

Answer 1

`f^-1(x)=-lnx`

Explanation:

`y=e^-x`

Switch the `x` and `y` and solve for `y`.

`x=e^-y`

To get the `-y` out of the exponent, take the natural logarithm of both sides. The logarithm is the inverse function of exponentiation, so it will undo the `e`.

`lnx=-y`

`y=-lnx`

In function notation:

`f^-1(x)=-lnx`

Answer 2

`y=-ln(x)`

`color(blue)(color(white)(....)"See explanation and graph")`

Explanation:

Let `y=e^(-x)`

Take logs both sides

`ln(y)=ln(e^(-x))`

`ln(y)=-xln(e)`

But `ln(e)=1` giving

`ln(y)=-x`

Swap the variable letters giving

`ln(x)=-y`

Multiply by (-1) giving

`y=-ln(x)`

`color(blue)("Something to think about!")`

The inverse function is a reflection of the original equation about the line `y= x`

In this case you have: